The Most Likely Transition Pathfor a Class of Distribution-Dependent Stochastic Systems

Wei Wei1,†, Jianyu Hu1,∗ and Jinqiao Duan2,‡

1Center for Mathematical Sciences, Huazhong University of Science and Technology,

Wuhan, Hubei 430074, China.

2Departments of Applied Mathematics & Physics, Illinois Institute of Technology, Chicago, IL 60616, USA.

Abstract

Distribution-dependent stochastic dynamical systems arise widely in engineering and science.We consider a class of such systems which model the limit behaviors of interacting particles mov-ing in a vector field with random fluctuations. We aim to examine the most likely transitionpath between equilibrium stable states of the vector field. In the small noise regime, we findthat the rate function (or action functional) does not involve with the solution of the skeletonequation, which describes unperturbed deterministic flow of the vector field shifted by the in-teraction at zero distance. As a result, we are led to study the most likely transition path for astochastic differential equation without distribution-dependency. This enables the computationof the most likely transition path for these distribution-dependent stochastic dynamical systemsby the adaptive minimum action method and we illustrate our approach in two examples.

Keywords and Phrases: Large deviations, interacting particles, adaptive minimum actionmethod, distribution-dependent stochastic dynamics, the most likely transition path, McKean-Vlasov stochastic systems

Lead paragraph:Large systems of interacting particles in vector fields are models in physics, bio-

sciences, social sciences and other fields. As the number of particles in such a systemincreases, the corresponding mean-field limit equation may be distribution dependent,such as the McKean-Vlasov stochastic system. By the recent large deviations resultson the distribution-dependent stochastic systems, we note that the rate function (oraction functional), which characterizes the probability of rare events, usually involveswith ‘modified deterministic flow’ (the flow of the vector field shifted by the interactionat zero distance). However, when we examine the most likely transition path betweenequilibrium stable states of the vector field, we find that the rate function does notinvolve with this modified deterministic flow. Consequently, we are able to computethe most likely transition path of a distribution-dependent system via a stochasticdifferential equation without distribution-dependency. We calculate the most likelytransition path by the adaptive minimum action method. As an illustration, we com-pute the most likely transition path for a distribution-dependent system for variousinteractions.

†Email: weiw16@hust.edu.cn∗Email: jianyuhu@hust.edu.cn‡Email: duan@iit.edu∗is the corresponding author

1

arX

iv:2

111.

0603

0v1

[m

ath.

NA

] 1

1 N

ov 2

021

Introduction

Large systems of interacting particles are now quite common in many fields. Particles canrepresent ions and electrons in plasmas in the context of physics [11]. In the bio-science, they canmodel the motion of cells or bacteria and in economics or social sciences, they represent “agents”[9, 10]. But they are analytically complex and computationally expensive as the number of particlesis very large. Instead, a corresponding mean-field limit equation is often an alternative model toinvestigate. There is a rich literature in deriving such mean-field limit equations; see, for example,[5, 6, 7]. Interactions in such equations often appear as the distribution in the drift term, which makethem known as the distribution-dependent equations. The McKean-Vlasov stochastic differentialequation was first studied in [8] and is a widely applicable one among these distribution-dependentequations.

The motion of the particles modeled by the McKean-Vlasov stochastic differential equationcan be simply described as follows: particles interacting with each other wander under randomfluctuations in a landscape, which is formed by a vector field. We are interested in how such aparticle transfers from one equilibrium stable state of the landscape to another equilibrium stablestate. Large deviations theory is a useful tool to deal with this kind of problem. Roughly speaking,the large deviations theory characterizes how small the probability of a rare event is via a ratefunction. It also extends the concept of potential to non-gradient system as quasi-potential, which iswidely used in investigating the long time behaviors of a system under small random perturbations.More details of the large deviations theory may be found in [2, 3].

There are many large deviations results investigating the McKean–Vlasov stochastic differentialequations. Herrmann et al. [4] derived a large deviation result for a one-dimensional McKean-Vlasovstochastic equation. Based on this, they investigated the influence of interactions on exit times andexit locations. Recently, Reis et al. [1] obtained large deviations results for a more general type ofn-dimensional McKean-Vlasov stochastic equations with Gaussian noise. Liu et al.[12] extendedthe large deviations results to the McKean–Vlasov stochastic equations with jumps. Although theseequations contain the distribution of the solution in the drift term, the rate functions they obtainedare simplified to only including a Dirac measure. Therefore, numerical methods built on the largedeviations theory, can be applied to investigate the dynamical behaviors of such McKean-Vlasovstochastic equations.

The minimum action method(MAM) is a widely used numerical method in finding the mostlikely transition path. It was proposed to compute the most likely transition path for stochasticdifferential equations with multiple metastable states [16]. The basic idea of MAM is to findthe minimizer of the associated rate function and take this minimizer as the desired most likelytransition path. MAM was improved by a reparametrization [13] to a fixed finite time scale andnumerically solving an Euler-Lagrange equation to find such a minimizer. A moving mesh methodwas used in [14, 15] to enhance the performance of MAM. Authors in [17] employed finite elementsapproximation to find the minimizer of the associated rate function.

In this paper, we calculate the rate functions for a class of McKean–Vlasov stochastic differentialequations. We find that the rate function (or action functional), which characterizes the probabilityof rare events, usually involves with a ‘modified deterministic flow’, which is the flow of the vectorfield shifted by the interaction at zero distance. When we examine the most likely transition pathbetween equilibrium stable states of the vector field, we find that the rate function does not involvewith this modified deterministic flow. An adaptive minimum action method [14, 15] is used tocalculate the most likely transition path.

The paper is arranged as follows. We briefly introduce the background of the McKean–Vlasovstochastic equation and the associated large deviations theory in the first section. Next, we analyze

2

the transition behaviors of particles modeled by the McKean–Vlasov stochastic equation. Finally,we present the most likely transition paths calculated by an adaptive minimum action method.

Backgrounds

We consider the following McKean-Vlasov equation with Xε0 = X0,

dXεt = V (Xε

t ) dt− F ∗ uεt(Xεt )dt+

√εdBt, (1)

where uεt stands for the law of Xεt . The symbol “∗” is the usual convolution operator that

F ∗ uεs(Xεs(ω0)) =

∫RF (Xε

s(ω0)− x)duεs(dx)

=

∫ω∈Ω

F (Xεs(ω0)−Xε

s(ω)) dP(ω) = E (F (Xεs(ω0)−Xε

s)) .

Equation (1) can be regarded as the mean field limit of the following system of many particles asN →∞, which is known as the propagation of chaos.

Let Xεi,N (0) = X0 and

dXεi,N (t) =

√εdBi,N (t) + V

(Xεi,N (t)

)dt− 1

N

N∑j=1

F(Xεi,N (t)−Xε

j,N (t))dt, (2)

where the Brownian motions Bi,N (t) are independent and F represents the interaction betweenparticles.

The solution Xεi,N is regarded as the motion of the N -th particle in a random environment that

interacts with all other particles therein. The solution Xε can be seen as the motion of a particlein a macroscopic system as N in equation (2) is the same magnitude as Avogadro’s constant. Themotion of Xε is generated by the vector field V , the interactions F and the random fluctuation Bt.

We are interested in the transition behaviors of Xε when the vector field V admits multipleequilibrium states, that is V = 0 at these states. Will these states become the metastable states ofXε or will the interaction F creates a different metastable state? How the most likely transitionpath looks like? The large deviations theory has shown its power in tackling such kind of problems.We will briefly introduce the large deviations theory in the following subsection.

The large deviations theory

Let C([0, T ]) denote the space of continuous functions on the interval [0, T ] taking values inRd, equipped with the uniform norm. Roughly speaking, if the solution Xε to (1) satisfies a largedeviation principle with a rate function IT on C([0, T ]), we can estimate the probability that Xε

stays in a neighborhood of a path ϕ as follows,

P‖Xε − ϕ‖ < δ ∼ exp

(−1

εIT (ϕ)

), as ε→ 0. (3)

The above asymptotic results can be used to estimate the probability of events associated with Xε

as ε tends to 0. To find the most likely transition path of Xε from point x1 to point x2 within timeT , we only need to solve the following constrained minimization problem. That is

− IT (ϕ∗) = −minϕIT (ϕ), (4)

3

where the minimization is constrained by

ϕ(0) = x1, ϕ(T ) = x2, ϕ ∈ C([0, T ]). (5)

The function ϕ∗T gives the most likely transition path within time T . The infimum value of equation4 over T ∈ [0,∞] is called quasi-potential V (x1, x2) between x1 and x2, that is

V (x1, x2) = infT>0

infϕ∈Cx2

x1([0,T ])

IT (ϕ), (6)

where Cx2x1 (0, T ) is the space of all absolutely continuous functions that start at x2 and end at x2.

Suppose that there exists a path ϕ and time T ∈ [0,∞] satisfying,

V (x1, x2) = IT (ϕ).

The path ϕ is the most likely transition path between point x1 and x2 that we are looking for.As is shown above, rate function governs the dynamical behavior of a stochastic system in the

context of the large deviations theory. With results derived in [1], we are able to calculate the ratefunction IT for our McKean–Vlasov equation (1).

Suppose the coefficient b(x, t, µt) = V (x) − F ∗ µt(x) satisfies [1, Assumption H2], where µt isthe law of a random variable. Let η be the solution to the following ordinary differential equation,

η(t) = V (η(t))− F ∗ δη(t)(η(t)) = V (η(t))− F (0), η(0) = x0. (7)

Equation (7) is often regarded as the skeleton equation. According to [1, Theorem 4.4], we knowthat Xε satisfies a large deviation principle in C([0, T ]) equipped with the topology of the uniformnorm with the good rate function

Ix0T (ϕ) =1

2

∫ T

0|ϕ(s)− V (ϕ(s)) + F (ϕ(s)− η(s))|2 ds, (8)

where ϕ is an absolute continuous function starting at x0. Xεt shares the same rate function with the

following corresponding stochastic differential equation if it satisfies the large deviations principle,

dZεt = (V (Zεt )− F (Zεt − η(t))) dt+√εdBt. (9)

Note that the drift term b may involve time t explicitly. We end this subsection with thecalculation of the rate functions for both one-dimensional and two-dimensional examples.

Example 1 (One-dimensional examples). We take V (x) = F (x) = x, and equation (1) reduces to

dXεt = E(Xε

t )dt+√εdBt. (10)

We take V (x) = −x+ x3 and F (x) = −x, and the skeleton equation (7) becomes

η = η − η3.

This system has two equilibrium stable states −1, 1 and an equilibrium unstable state 0. The ratefunction is

Ix(ϕ) =1

2

∫ T

0|ϕ(s) + ϕ3(s)− η(s)|2ds.

The corresponding stochastic differential equation (9) becomes

dZεt =(−(Zεt )

3 + η(t))dt+

√εdBt.

4

Example 2 (The Maier-Stein model with interaction). Take

V (u, v) =

(u− u3 − βuv2

−(1 + u2

)v

)and F (u, v) =

(u−v

).

The skeleton equation (7) becomes,η(t) = V (η(t)).

Then the corresponding stochastic differential equation (9) is

dZεt = b(Zεt , t)dt+√εdBt,

where

b(u, v, t) =

(−u3 − βuv2 − η1(t)−u2v + η2(t)

)and η(t) =

(η1(t)η2(t)

).

The rate function for this equation is

Ix(ϕ) =1

2

∫ T

0

∣∣∣ϕ(s)− b(ϕ(s), s)∣∣∣2 ds.

Dynamical behaviors of the McKean–Vlasov stochastic differential equation

We are interested in the transition behavior of the McKean–Vlasov equation between the equi-librium stable states of the vector field V . Based on the large deviation theory, we will analyze thedynamical behavior of a class of McKean–Vlasov stochastic differential equations in this subsection.

Let x1 and x2 be two equilibrium stable states of (11), that is V (x1) = V (x2) = 0. If theinteraction F is 0 at point 0, then equation (7) starting at x1 reduces to

η(t) = V (η(t)), η(0) = x1. (11)

When we look for the most likely transition path from x1 to x2, the solution η stays at x1. Thenthe rate function in this case is

Ix1T (ϕ) =1

2

∫ T

0|ϕ(s)− V (ϕ(s)) + F (ϕ(s)− x1)|2 ds =

∫ T

0|ϕ− b(ϕ(s))|2ds. (12)

The corresponding stochastic differential equation (9) then becomes

dZεt = (V (Zεt )− F (Zεt − x1)) dt+√εdBt = b(Zεt )dt+

√εdBt, Zε0 = x1. (13)

For the solution ξ to the following equation,

ξ(t) = V (ξ(t))− F (ξ(t)− x1), (14)

we know that Ix1T (ξ) = 0, which is the minimum value of the rate function Ix1T . So the quasi-potential between every two different points on the path ξ is 0. Then the trajectories of ξ startingat different points form a set of equipotential lines. In addition, we assume that the equilibriumstable states x1 and x2 are also asymptotic stable states of system (11). Then for the followingsystem

dY εt = V (Y ε

t )dt+√εdBt, (15)

x1 and x2 become two metastable states. It means that Y ε starting near one of these two states,will stay a fairly long time near this state. But for the solution Zε to the corresponding stochastic

5

differential equation (11), x1 and x2 may not be metastable states due to the presence of theinteraction term F . Instead, we need to find the equilibrium states of equation (14) to determinethe metastable states of Zε. As is shown in next section, the most likely transition path will passby a saddle point and be attracted to a “interaction-generated” metastable state of Zε.

Note that the integrand b does not involve time t explicitly. We can then use a reparametrizationmethod to find the most likely transition path over time T ∈ [0,∞], which is known as the geometricminimum action method [13]. For the cases that the drift term b in equation (9) involves time texplicitly, the adaptive minimum action method [14] can be used to compute the most likelytransition path.

Compute the most likely transition path for the McKean-Vlasovstochastic differential equation

In this section, we will present the results generated by the adaptive minimum action method[14], which is widely used in computing the most likely transition path in the context of largedeviations theory.

The idea of the adaptive minimum action method(aMAM) is to find the minimizer of (6), bynumerically solving an associated Euler-Lagrange equation with a fixed time T . The moving meshstrategy is used to choose a proper mesh grid via a monitor function in every iteration. Theassociated Euler-Lagrange equation is then solved on such mesh grids. Choose a large enough Tand the numerical solution is the most likely transition path we are looking for.

We will compute the most likely transition path for the Maier-Stein model with different inter-action terms. That is

dXεt = V (Xε

t ) dt− F ∗ uεt(Xεt )dt+

√εdBt, Xε

0 = (−1, 0), (16)

where

V (u, v) =

(u− u3 − βuv2

−(1 + u2

)v

), β = 10.

In Figure 1, we calculate the most likely transition path connecting two equilibrium stable states(−1, 0) and (1, 0), under the interaction F = FL, which is

FL(u, v) =

(u−v

).

The corresponding stochastic differential equation (13) then becomes

dZεt = bL(Zεt )dt+√εdBt, Zε0 = (−1, 0),

where

bL(u, v) =

(−u3 − βuv2 + 1

−u2v

).

The rate function is

IL(ϕ) =

∫ T

0|ϕ− bL(ϕ(s))|2ds.

There is only one asymptotic stable point (−1, 0) for the following equation,

ξL(t) = bL(ξL(t)).

6

Figure 1: The most likely transition path for the Maier-Stein model with interaction FL. Theorange line is the most likely transition path connecting points (−1, 0) and (1, 0). Grey lines arethe equipotential lines of the associated quasi-potential. We set the number of points N = 200,iteration K = 5000 and time T = 20. The initial path is chosen as y = −0.5x2 + 0.5.

It is also the metastable point of the corresponding stochastic differential equation. The grey linesin Figure 1 are the orbits of ξL starting at different points and these form the equipotential linesof the quasi-potential. The interaction F eliminates the metastable point (1, 0). The most likelytransition path in this case is actually the most likely exit path from (−1, 0) to (1, 0).

In Figure 2, we calculate the most likely transition path under the interaction described by amodified Biot-Savart kernel which is not singular at the origin. Precisely, we take F = FBS to be

FBS(u, v) =1

2π

( −vu2+v2+δ

uu2+v2+δ

), δ = 0.01.

The corresponding stochastic differential equation (13) then becomes

dZεt = bBS(Zεt )dt+√εdBt, Zε0 = (−1, 0),

where

bBS(u, v) =

(u− u3 − βuv2 + v

2π((u+1)2+v2+δ)

−(1 + u2)v − u+12π((u+1)2+v2+δ)

).

The rate function is

IBS(ϕ) =

∫ T

0|ϕ− bBS(ϕ(s))|2ds.

For the presence of the interaction term FBS , there are two asymptotic stable points x1 = (−1, 0),x2 ≈ (1.14801, 0.0318893) and a saddle point x∗ ≈ (0.009145,−0.152703) for the following equation,

ξBS(t) = bBS(ξBS(t)). (17)

7

Figure 2: The most likely transition path for the Maier-Stein model with interaction FBS . Theorange line is the most likely transition path connecting points (−1, 0) and (1, 0). Grey lines arethe equipotential lines of the associated quasi-potential. The dark grey point is the saddle point ofequation (17). We set the number of points N = 400, iteration K = 10000 and time T = 40. Theinitial path is chosen as y = −0.5x2 + 0.5.

The metastable points for the corresponding stochastic differential equation are then x1 and x2.The grey lines in Figure 2 are the orbits of ξBS starting at different points and these form theequipotential lines of the quasi-potential. The saddle point x∗ is the dark grey point in Figure 2.The interaction F transports the saddle point (0, 0) and stable point (1, 0) to x∗ and x2 respectively.Then the most likely transition path will pass by the saddle point x∗ and be attracted to theneighborhood of x2 as is shown in Figure 2.

Conclusion

We have investigated the most likely transition path for a class of distribution-dependentstochastic dynamical systems. In the small noise regime, this problem turns out to be equivalentto computing this transition path for a different stochastic dynamical system without distribution-dependency. Therefore, we can apply the adaptive minimum action method to calculate the mostlikely transition paths, as demonstrated in several examples.

Acknowledgements

This work was partly supported by the NSFC grants 11771449.

8

Data Availability

The data that support the findings of this study are openly available in GitHub: https:

//github.com/JayWeiess/MVSDE-aMAM

References

[1] G. Reis, W. Salkeld and J. Tugaut. Freidlin–Wentzell LDP in path space for McKean–Vlasovequations and the functional iterated logarithm law. The Annals of Applied Probability, 29:1487– 1540, 2019

[2] A. Dembod and O. Zeltouni. Large Deviations Techniques and Applications. Springer, Berlin,Heidelberg, 2010.

[3] M. I. Freidlin and A. D. Wentzell. Random Perturbations of Dynamical Systems. Springer,Berlin, Heidelberg, 2012

[4] S. Herrmann, P. Imkeller and D. Peithmann. Large deviations and a Kramers’ type law forself-stabilizing diffusions. The Annals of Applied Probability, 18:1379 – 1423, 2008.

[5] F.Bolley, A. Guillin and C. Villani. Quantitative concentration inequalities for empirical mea-sures on non-compact spaces. Probability Theory and Related Fields, 137:541 – 593, 2007.

[6] F. Cucker and S. Smale. Emergent behavior in flocks. IEEE Transactions on AutomaticControl, 52:852– 862, 2007.

[7] P.E. Jabin. A review for the mean field limit for Vlasov equations. Kinetic and Related Models,7:661– 711, 2014.

[8] H. P. McKean. A class of Markov processes associated with nonlinear parabolic equations.Proceedings of the National Academy of Sciences, 56: 1907 – 1911, 1966.

[9] R. Carmonaand F. Delarue. Probabilistic Theory of Mean Field Games with Applications I.Springer, Cham, 2017.

[10] R. Carmonaand F. Delarue. Probabilistic Theory of Mean Field Games with Applications II.Springer, Cham, 2017.

[11] S. B. Pope. Turbulent Flows. Cambridge University Press, 2000.

[12] W. Liu, Y. Song, J. Zhai and T. Zhang. Large and moderate deviation principles for McKean-Vlasov SDEs with jumps. arXiv preprint arXiv:2011.08403 2020

[13] M. Heymann and E. Vanden-Eijnden. The geometric minimum action method: A least actionprinciple on the space of curves. Communications on Pure and Applied Mathematics: AJournal Issued by the Courant Institute of Mathematical Sciences, 61: 1052–1117, 2008.

[14] Y. Sun and X. Zhou. An improved adaptive minimum action method for the calculationof transition path in non-gradient systems. Communications in Computational Physics, 24:44–68, 2018.

[15] X. Zhou, W. Ren and W. E. Adaptive minimum action method for the study of rare events.The Journal of Chemical Physics, 128: 104–111, 2008.

9

[16] W. E, W. Ren and E. Vanden-Eijnden. Minimum action method for the study of rare events.Communications on Pure and Applied Mathematics, 57: 637–656, 2004.

[17] X. Wan, H. Yu and J. Zhai. Convergence analysis of a finite element approximation of minimumaction methods. SIAM Journal on Numerical Analysis, 56: 1597–1620, 2018.

10